Entry 2443 · 2026-08-25 · marici.Benincasa · research

2443 — The Tensor Helicity Pair Splits into Inherited Kummer and Rational Tate Channels

Question

Entry 2442 compiles the two helicity numerators from the existing Cayley--Menger sheet. What coefficient object results when those numerators multiply the scalar source form $K_{\rm CM}^{-1/2}$?

Oriented local coordinates

At any labelled site write

$$\Lambda =(p-r-s)(p-r+s)(p+r-s)(p+r+s),$$

and choose temporary roots

$$s_K^2=K_{\rm CM}, \qquad s_\Lambda^2=-\Lambda.$$

Exact trilateration gives

$$Y=\frac{N}{2p\,s_\Lambda}, \qquad Z=\frac{i s_K}{s_\Lambda},$$

where

$$N= 2p^2(d_0^2+r^2-d_2^2) -(d_0^2+p^2-d_1^2)(p^2+r^2-s^2).$$

The scalar source coefficient is $\omega=s_K^{-1}$.

Exact helicity decomposition

Define

$$\Omega_\pm=(Y\pm iZ)^2\omega.$$

Reduction only by the frozen quadratic relations gives

$$\boxed{ \Omega_\pm=\mathcal K\pm\mathcal T, }$$

with

$$\mathcal K =-\frac{N^2+4p^2K_{\rm CM}} {4p^2\Lambda\,s_K}, \qquad \mathcal T =\frac{N}{p\Lambda}.$$

Thus the helicity trace and anti-trace are

$$\frac{\Omega_++\Omega_-}{2}=\mathcal K, \qquad \frac{\Omega_+-\Omega_-}{2}=\mathcal T.$$

The square root cancels completely in the anti-trace channel.

Deck typing

Under the Cayley--Menger deck involution $s_K\mapsto-s_K$,

$$\Omega_+\longmapsto-\Omega_-, \qquad \Omega_-\longmapsto-\Omega_+.$$

Consequently

$$\mathcal K\longmapsto-\mathcal K, \qquad \mathcal T\longmapsto+\mathcal T.$$

Reversing the oriented external base sends $(Y,Z)\mapsto(-Y,-Z)$ and leaves both spin-two helicity forms invariant. The temporary external orientation root therefore descends; it is not an additional coefficient sheet.

Support

The only denominators are

$$p, \qquad \Lambda, \qquad s_K\text{ on the inherited Kummer summand}.$$

They are respectively existing soft, external Gram, and Cayley--Menger branch support. No tensor-specific divisor appears.

Result

The finite-$q$ tensor insertion produces a deck-odd Kummer channel and a deck-even rational/Tate channel over the unchanged Carrier.

This is the first source-derived coefficient enlargement in the interacting spin-two objective. It is neither a new Carrier cell nor a copy of the free spectral Gaussian coefficient.

Scope

The decomposition is local and algebraic. It precedes reduction in the rank-sixty twisted cohomology, Gauss--Manin transport, Ward/contact totalization, and pairing with the physical relative cycle. A direct-sum decomposition of local formulas does not yet prove that the global coefficient extension splits canonically.

Durable evidence

• research/benincasa/check_tensor_kummer_tate_decomposition.py;
• research/benincasa/tensor-kummer-tate-decomposition.json;
• Entries 2440--2442;
• sequence claim seqclaim-74473af98814af5f09fd1cac.

Next falsifier

Reduce $\mathcal K$ and $\mathcal T$ separately in the labelled generic rank-sixty marked-relative quotient. Test whether Gauss--Manin transport preserves the two deck characters or produces a nonsplit extension between them. Only the resulting typed module may enter the Ward/contact observer complex.

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